When we practice a motor task, we can do it better the next time we revisit it. How is this accomplished? The basic assumption in neuroscience has been that this phenomenon, called 'savings', occurs because when we revisit the task, the brain recalls the motor commands that it had previously learned. In this view, motor memory is a memory of motor commands. In contrast to these views, here we propose that motor memory includes a memory of errors, i.e., when we are better at a task, it is often not because we remember the motor commands that we learned before, but because we remember the errors that we have seen before. We propose that this counter-intuitive and previously unknown form of memory, a memory of errors, can account for a very large body of puzzling and unexplained experimental data in the motor learning literature. In this proposal, we begin with a puzzle: in a motor learning task, humans are able to modulate how much they learn from a given error. In some conditions, they learn a large amount, but in other conditions they learn only a small amount. That is, the brain selects how much it is willing to learn from error. To understand the rules that govern control of error-sensitivity, we manipulate the history of errors that subjects experience, and find that positive autocorrelations of errors up-regulate sensitivity, whereas negative autocorrelations down-regulate sensitivity, but only at the specific errors that were actually experienced. That is, experience of an error produces a change in motor commands via errors-sensitivity, which in turn depends on the history of past errors, allowing the brain to respond differently to an error that was experienced before. We formulate this idea with a set of mathematical equations that extend the current framework of learning and show that this new set of equations, representing memory of errors, seamlessly connects a large body of puzzling data. Building on these preliminary results, we propose two groups of experiments to test the foundations of this new idea. The first set of experiments test the prediction that correlations of past errors will modulate error-sensitivity, and that this modulation will be local to the specific errors that were experience. The second set of experiments test the prediction that modulation of error-sensitivity via memory of errors is the basis for the phenomenon of saving. From a clinical standpoint, understanding error-sensitivity is important as it directly affects motor rehabilitation for neuro-trauma or disease. Our theory provides a recipe to modulate error-sensitivity, which should produce faster adaptation, potentially affecting the duration of rehabilitation. In addition, understanding the relationship between error sensitivity and savings may provide useful clues regarding how to effectively apply rehabilitation techniques to promote faster re-learning outside the clinic.